On degree anti-Ramsey numbers

TL;DR

This paper investigates the degree anti-Ramsey number of graphs, establishing bounds, exact values for forests, and near-optimal bounds for cycles, using combinatorial and topological tools.

Contribution

It provides a general upper bound on degree anti-Ramsey numbers, exact values for forests, and near-optimal bounds for cycles, advancing understanding of graph coloring properties.

Findings

Exact degree anti-Ramsey numbers for forests.

Upper bounds on degree anti-Ramsey numbers for cycles.

A general upper bound on degree anti-Ramsey numbers.

Abstract

The degree anti-Ramsey number $AR_d(H)$ of a graph $H$ is the smallest integer $k$ for which there exists a graph $G$ with maximum degree at most $k$ such that any proper edge colouring of $G$ yields a rainbow copy of $H$. In this paper we prove a general upper bound on degree anti-Ramsey numbers, determine the precise value of the degree anti-Ramsey number of any forest, and prove an upper bound on the degree anti-Ramsey numbers of cycles of any length which is best possible up to a multiplicative factor of $2$. Our proofs involve a variety of tools, including a classical result of Bollob\'as concerning cross intersecting families and a topological version of Hall's Theorem due to Aharoni, Berger and Meshulam.